Subject: Question about manifolds
Date: Tue, 19 Feb 2002 14:39:14 -0500
From: Gregory Arone
To: dmd1@lehigh.edu
Dear Don
Could you post this for me?
Thanks, Greg.
Let M be a manifold, N an embedded submanifold
(for simplicity, manifold = compact manifold without
boundary). Consider the space M-N (the complement
of N in M). I would like to "compactify" this space, without
changing the homotopy type, by "glueing" the sphere
bundle of the normal bundle of N to M-N. Such
a compactification, once constructed, will be denoted
cl(M-N).
One way to construct cl(M-N) is to choose an open tubular
neighborhood of N in M, and take the complement of
this neighborhood. My question is: is there a more
canonical way to do it?
Ideally, I would like a construction for cl(M-N) that does
not involve a choice of anything like a Riemannian
metric, nor a choice of a complement of the tangent
bundle of N in the tangent bundle of M. The construction
should be as functorial as one could reasonably expect.
E.g., an embedding f:M--> M' would induce a map
cl(M-N)-->cl(M'-f(N))
To summarize, my questions are:
Is such a construction possible?
Is it described in the literature?
With many thanks
Greg